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Advances in Environmental Biology, 8(11) Special 2014, Pages: 1330-1337
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Advances in Environmental Biology
ISSN-1995-0756
EISSN-1998-1066
Journal home page: http://www.aensiweb.com/AEB/
Analyzing Generalized Matching Point Technique (GMT) in Order to study
Electromagnetic Interaction with Nanoparicles
Jafari Marjan, Jalali Tahmine, Safaie Amin, Mohammadi Eslami Ahmad
Physics department in Persian Gulf University
ARTICLE INFO
Article history:
Received 15 April 2014
Received in revised form 22 May
2014
Accepted 25 May 2014
Available online 15 June 2014
Keywords:
generalized matching point technique
(GMT), nanojet, localized surface
plasmon (LSP)
ABSTRACT
In this paper, semi-analytical generalized matching point technique (GMT) is
investigated in order to study the interaction between electromagnetic wave and
nanostructures. GMT code is developed in MATLAB software. To validate the code,
both analytical Mie theory and semi-analytical GMT method are applied
simultaneously for simulating scattered field from dielectric cylinder and a gold
nanowire in vacuum which are both illuminated by a TM polarized plane wave. It can
be seen that the agreement between both techniques is excellent. It is turned out that
nanojet in dielectric micro-cylinder and localized surface plasmon (LSP) in gold
nanowire are formed which both Mie and GMT techniques predict the same result.
© 2014 AENSI Publisher All rights reserved.
To Cite This Article: Jafari Marjan, Jalali Tahmine, Safaie Amin, Mohammadi Eslami Ahmad, Analyzing Generalized Matching Point
Technique (GMT) in Order to study Electromagnetic Interaction with Nanoparicles. Adv. Environ. Biol., 8(11), 1330-1337, 2014
INTRODUCTION
Analyzing the interaction of electromagnetic waves with nano cylindrical and spherical particles by using
different numerical techniques has been the subject of intense investigation and research, recently. Employing
an optimum method by regarding the condition governing these sorts of problems is a really crucial subject.
Generalized Mie theory has been introduced as a regular analytical solution for studying light scattering with
spherical and cylindrical particles [2,3]. Since generalized Mie theory is just capable of computing scattered
field from a single sphere or a single cylinder, so for simulating more complex problems and also for achieving
more realistic results, various numerical methods are being applied. For instance, finite differential time domain
(FDTD) and finite element (FEM) techniques can be mentioned. FDTD method is a high-resolution code which
is performed in time domain. Time consuming and instability are regarded as the most important disadvantages
of this numerical technique. Moreover, it is required to mesh the simulation space as fine as possible in order to
obtain reliable results that need really high computation resources. Thus, employing boundary methods in
frequency domain may be more convenient, since only interfaces need to be descritized. The generalized
matching point technique (GMT) is a good semi-analytical candidate for satisfying all computational
requirements. GMT is a technique which works in frequency domain and as it is claimed to be semi-analytical
solution so it possesses a very high accuracy and precision that is valuable when nanostructures are explored.
The other strong feature which makes this method unique is that GMT is fast in manipulating heavy computing.
It should be pointed out that the optimum location of multipoles is explored by trial and error. Hence, arranging
them in a way that the residual is minimized is an extremely difficult task.
Photonic nanojets appear on the shadow side of the illuminated dielectric particle [11]. These particles
create narrow and high intensity beams undergo the diffraction limit in the near field of the particle surface.
With advent of achieving the undergo diffraction limit knowledge, lots of applications have been turned out
including scanning near-field optical microscopy (SNOM) [12], high resolution lithography, advanced optical
(and magnetic) data storage schemes, sensing and metrology, optical trapping, nano-patterning and immersion
lens microscopy, Raman spectroscopy and for fluorescence imaging of living cells such as single molecules
[4,6]. On the other hand, metallic nanostructures have broad application in nanoptic due to surface plasmon and
localized surface plasmon phenomena [7]. Hence, with respect to wide practical domain, studying and
investigating an optimum method for modeling these sorts of structures is absolutely a vital task [11].
In this paper, we present the generalized matching point technique initially. Afterwards, this numerical
method is adapted for simulating the scattered field outside a dielectric micro-cylinder and a gold nanowire, as
Corresponding Author: Jafari Marjan, Physics department in Persian Gulf University
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Advances in Environmental Biology, 8(11) Special 2014, Pages: 1330-1337
well. Finally, to validate the developed GMT code, the simulation results are compared with analytical Mie
theory and it would be shown that both techniques reach to the same results.
GMT theory:
The generalized matching point technique was developed in Swiss Federal Institute of Technology by
Hafner in 1980. The GMT is a semi-analytic boundary method working in the frequency domain for calculating
electromagnetic fields via solving Maxwell equations in linear, isotropic and homogenous media and provide
highly accurate results. The GMT has originated from the analytical Mie theory, circular harmonic analysis and
point matching. Hence, the GMT method is claimed to be semi-analytical because the approximation to the
actual field analytically satisfies the Maxwell differential equations, while the algebraic boundary conditions are
approximately fulfilled at matching points.[7]
The GMT technique is based on this reality which every arbitrary electromagnetic field in the medium can
be expanded by harmonic basis functions that are defined as orthonormal set. It should be concerned that
harmonic basis functions must satisfy 2D Helmholtz equation, essentially.
Basis functions and field expansions:
A peculiarity of GMT is the high degree of freedom in selection of harmonic basis functions, in particular
the multipolar functions. Modeling an electrodynamics problem with multipolar sources requires physical
concepts. As it was mentioned before, fields can be simulated as a superposition one or more multipolar sources
everywhere. If the point, that the fields are to be calculated there, is located inside the domain where the origins
of multipoles have been placed, to prevent singularity, Bessel functions would be employed. Likewise, if the
fields computing point is located outside the region of multipole origins, Henkel functions would be substituted
instead of multipolar functions, since Henkel functions are singular in the origin whereas do not vanish at
infinity. In the multiple multipole program (MMP), which is a special kind of GMT method, Henkel functions
are adapted for field expansion [8]. Therefore, according to figure (1), in order to model the electromagnetic
field in domain 𝐷𝑖 the origins of multipolar functions have to be located in region 𝐷𝑖 and similarly, to model the
fields in domain 𝐷𝑗 multipolar function origins have to be placed in region 𝐷𝑖 .
Fig. 1: Multipole expansion distribution. For modeling the electromagnetic field in domain 𝐷𝑖 the origins of
multipolar functions have to be located in region 𝐷𝑖 and similarly, to model the fields in domain 𝐷𝑗
multipolar function origins have to be placed in region 𝐷𝑖 . The o’s and ×’s denote the origins of
multipole in domain 𝐷𝑖 and 𝐷𝑗 , respectively.
In GMT method, to model 2D electrodynamics problems such as problems with cylindrical structure (i.e.
structures that are invariant along one axis (cylinder axis)) the knowledge of longitudinal components of the
D i ,D j
electric and magnetic fields {Ez
magnetic field
𝐷𝑖 ,𝐷𝑗
𝐻𝑇
in
D i ,D j
, Hz
D i ,D j
}, is sufficient to derive the transverse electric field ET
both domains 𝐷𝑖 and 𝐷𝑗 . Therefore, according to figure (1), when an electromagnetic
D i ,D j
emission interacts with a particle φDexci , the longitudinal components of both Ez
a following form in a transverse plane:
D i ,D j
Ez
D i ,D j
Hz
D i ,D j
KT
Nl
n=0
Nl
n=0
and
D i ,D j
=
L
l=1
=
L
l=1
=
εD i , D j μ D i , D j ω 2 − γ 2 =
AEn l Zn K T
AHn l Zn
D i ,D j
rl cos(nφl ) + BnEl Zn K T
D i ,D j
K T rl
cos(nφl ) +
BnHl Zn
εD i ,D j μD i ,D j ω2 − k 2z
rl
D i ,D j
K T rl
D i ,D j
and Hz
are formulated in
sin(nφl )
(1)
sin(nφl )
(2)
(3)
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s where l denotes the number of multipoles, n is the order of multipole, L and 𝑁𝑙 represent maximum number
and order of essential multipoles, respectively, rl , φl represent the polar coordinate of observing point from the
D
lth multipole view and Zn K T i rl is a determined cylindrical function, in which depending on multipole
distribution, can be substituted by Bessel or Henkel functions. Thus, if Henkel functions Hn are used as
cylindrical functions, the expansion would be called singular multipole since Henkel is singular in the origin, so
it is appropriate to be adapted for describing the scattered field. .AEn l , AHn l , BnEl , and BnHl are expansion
D
coefficient which have to be determined. In equation (3), K T i is the transverse wave number, εD i and μD i denote
permeability and permittivity coefficient in domain 𝐷𝑖 , respectively, and ∇T represents the gradient in XY
D i ,D j
plane. When both Ez
∇T +
D i ,D j 2
KT
D i ,D j
and Hz
satisfy the 2D Helmholtz equation in 𝐷𝑖 :
D i ,D j
Ez
D i ,D j
Hz
=0
(4)
then, the transverse components of electric and magnetic fields fulfill the differential Maxwell equations in
𝐷𝑖 :and are derived in the following form:
D i ,D j
ET
D i ,D j
HT
(r) =
(r) =
i
D i ,D j 2
KT
i
D i ,D j 2
KT
D i ,D j
k z ∇ T Ez
D i ,D j
k z ∇T Hz
D i ,D j
r − ω μD i ,D j (ez × ∇T )Hz
(5)
D i ,D j
r + ωεD i , D j (ez × ∇T )Ez
(6)
where ez is a unit vector along the Z axis.
Boundary conditions:
Therefore, according to figure (1) and with respect to the presented expansions, the total field in domain 𝐷𝑖
is given by
Di
i
φDapp
= φexc
+ φDscai
(7)
Di
where φapp
denotes the approximation to the actual field, term φDexci represents determined exciting field
,and φDscai is unknown scattered electromagnetic field in the corresponding domain and is replaced by proper
expansions given in equations (1), (2), (5), and (6). To determine the unknown coefficients based on media
properties and also polarization of incident radiation, the appropriate weighted boundary conditions are imposed
on the set of collocation points along the interface. So that, according to figure (2), the boundary of particle is
partitioned in some segments. Within GMT method, to impose boundary condition, a generalized point
matching (GPM) technique is utilized in order to minimize the weighted residual of electromagnetic field
continuity within a finite number of matching points on boundary of object between homogenous domains.
Thus, applying the proper continuity conditions on tangential and transverse components of electromagnetic
field along all of the matching points may lead to a matrix equation of type
(8)
β A αβ Fβ = Gα
where, Aαβ is a rectangular matrix, the vector Fβ includes the unknown coefficients and the vector Gα stems
for excitation. In order to reduce the numerical errors and also for more continuity on interface, the technique,
GMT method works with more equations (usually 2 or 3 times) than unknowns which is leading to an
overdetermined system, and would be solved numerically. Eventually, after determining unknown coefficients,
it is possible to compute field and also all other physical quantities such as pointing vector at every arbitrary
point.
Fig. 3: Partitioning the interface between domains 𝐷𝑖 and 𝐷𝑗 , to impose boundary conditions in GMT method
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2D GMT simulation:
In this paper, we report on the manipulation of GMT method for simulating scattered field from a dielectric
micro-cylinder and also a gold nanowire which are illuminated by a plane harmonic wave such a way that the
direction of the propagation is perpendicular to the cylindrical axis and it has TM polarization (i.e., the waves in
which their components 𝐻z and 𝐸T are not zero). Afterwards, we compare the computational results with
analytical Mie solution, as well. As the first attempt, we have to determine the optimum position of multipoles
due to the medium condition and also the geometry of relevant structure. Suppose a micro-cylinder in a way that
its axis is along the Z axis, as a result the symmetry axis would be Z axis. As it is shown in figure (3), the
problem possesses symmetry. So, just one multipole is sufficient for modeling scattered field and similarly one
for simulating internal field. The optimum location of both multipoles is in the center of circle (cylinder cross
section).
Fig. 3: 2D GMT simulation space geometry: the ’s denotes the location of matching points on cylinder
interface and ×’s and +’s represent multipole origins to model internal and scattered fields, respectively.
The properties of plane wave are indicated at the right side. Cylinder axis is along Z direction.
Since the origin of multipole modeling the internal field is inside the micro-cylinder, so, as it was
mentioned before, we employ Bessel function as cylindrical functions in the expansions represented for
electromagnetic field to avoid singularity in the center. Likewise, as the multipole origin, simulating scattered
fields is located outside the scattering region, so that, it is reasonable to apply Henkel function in the
corresponding expansion because Henkel functions are finite at infinity. Due to the polarization of incident
wave (TM polarization), it is just sufficient to impose tangential boundary condition on electric and magnetic
field components along all collocation points in order to determine the unknown coefficients. It is worth
mentioning that if the numbers of multipolar expansion terms applied for modeling scattered and internal fields
are less than an optimum value, so the convergence may not be achieved and the simulation results would be far
from reality.
RESULTS AND DISCUSSIONS
In this framework, to study the scattering problem we have developed the GMT code in MATLAB. A
dielectric cylinder with radius R=2μm is illuminated by a unit amplitude plane wave with operating wavelength
λ = 400𝑛𝑚 and also TM polarization. The refractive index of this micro-cylinder is n=1.45 and the
background medium is vacuum (𝑛𝑜𝑢𝑡 = 1). In order to validate the developed code, we compare the simulation
data of scattered field intensity with analytical Mie theory as it is depicted in figure (4).
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Fig. (4): A dielectric cylinder with radius R=2μm is illuminated by a unit amplitude plane wave with operating
wavelength λ = 400𝑛𝑚 and also TM polarization. The background medium is vacuum.
In the figure (4) it can be seen that agreement between both techniques (including all the points particularly
a maximum value of the scattered electric field intensity) is excellent and this indicates extreme convergence of
GMT code. In this computation, we take the number of matching points on the boundary in a way that the
outcome of imposing boundary conditions on them leads to numbers of equations which is two times more than
numbers of unknown factors. It should be noted that the numbers of matching points cannot be chosen more
than an optimum value due to the computation speed, on the other hand, as it was said before, the matching
points numbers should not be selected too small such a way that the numbers of unknown terms become more
than numbers of equations. In the corresponding dielectric micro-cylinder in order to calculate scattered field
using GMT method, we placed just one multipole of order 46 in the center of the cylinder and the intensity of
scattered Hz is depicted as a function of angles between 0 to 2π at distance twice the radius (4μm) in figure (4).
Fig. 5: 2D GMT simulation of scattered electric field intensity pattern from a dielectric cylinder with radius
R=2μm is illuminated by a unit amplitude plane wave with operating wavelength λ = 400𝑛𝑚 and also
TM polarization at distance twice the radius (4μm). The refractive index of this micro-cylinder is n=1.45
and the background medium is vacuum (𝑛𝑜𝑢𝑡 = 1).
Afterwards, we simulate the pattern of scattered electric field intensity at the distance twice the radius and
with the same mentioned medium properties. The results are displayed in figure (5). As it is observed in figure
(5), according to the represented concepts, nanonojet is provided just in the location where we expected, namely
at the left end of micro-cylinder and exact in the direction of incident radiation. This is additional reason of
GMT code validity and accuracy. Relative error graph of the respective computation is depicted in figure (6)
and it is approximately in the order of 10-4 which indicates high precision of GMT technique.
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Fig. 6: Relative error graph in GMT method.
Fig. 7: 2D GMT simulation of the scattered magnetic field in Z direction from a gold nano-cylinder with radius
R=150nm employing both Mie GMT methods. The background medium is vacuum (𝑛𝑜𝑢𝑡 = 1) and the
refractive index of gold for incident wavelength (λ=547nm) is 𝑛𝑖𝑛 = .44205 + 2.4313i
At the remainder of this paper, we investigate the simulation of interaction between electromagnetic field
and gold nanowire which is important due to frequent application in optoelectric and nanoptic fields. The
scattered magnetic field in Z direction from a gold nano-cylinder with radius R=150nm is presented. The
background medium is vacuum (𝑛𝑜𝑢𝑡 = 1) and the refractive index of gold for incident wavelength (λ=547nm)
is 𝑛𝑖𝑛 = .44205 + 2.4313i.
The exciting electric field is contained in XY plane (i.e., TM polarization). In figure (7), the intensity of
scattered Hz is plotted as a function of angles between 0 to 2π at distance twice the radius (300nm) employing
both Mie and GMT methods. Again due to the symmetry of the problem, the optimum place of multipole is in
the center and just one multipole is enough. As it is observed in figure (7), the scattered H z intensity of gold
nanowire under the mentioned physical properties, has a peak in angle π radian. It is deduced that a plane wave
with TM polarization and at operating wavelength λ=547nm excites localized surface plasmon (LSP) in the
relevant gold nanowire. To prove this claim, we compute the scattering cross section of gold nanowire in the
wavelength interval 200 to 1000nm using both Mie and GMT techniques. The results of calculations are
indicated in figure (8). As we expected, scattering cross section is maximum in λ=547nm and 𝑛𝑖𝑛 = .44205 +
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Advances in Environmental Biology, 8(11) Special 2014, Pages: 1330-1337
2.4313i . It just demonstrates this reality that LSPR is formed in gold nanowire in the corresponding
wavelength.
Fig. 8: Scattering cross section as a function of wavelength interval 200 to 1000nm using both Mie and GMT
techniques. LSP is occurred in the wavelength λ=547nm.
Conclusion:
In this paper, we employed analytical Mie theory for modeling scattered field from a dielectric microcylinder and a gold nanowire in order to examine the semi-analytical GMT technique. By comparing Mie and
GMT methods, we conclude that GMT technique as a semi-analytical method, compared with other numerical
methods, possesses high accuracy and essential precision for modeling nanoptic and optoelectronic problems.
Moreover, it is absolutely fast and has high convergence, as well. With regard to the performed simulations in
this investigation, it is deduced that in GMT method the best and the most possible accurate place of multipole
for modeling scattered field from a cylindrical structure is in the center of circle (cross section of cylinder) and
also only one multipole is sufficient for this kinds of structures. In order to validate, both GMT code and Mie
theory have been applied simultaneously, for simulating scattered field from a dielectric cylinder with refractive
index 𝑛𝑖𝑛 =1.45 and radius R=2μm in vacuum which is illuminated by a TM polarized plane wave with
operating wavelength λ=400nm and the outcome agreed with Mie theory greatly. In this dielectric microcylinder scattered Hz intensity was depicted as function of angles interval 0 to 2π at distance 2R and it was
observed that Hz intensity has its maximum value in π radian. Moreover, by plotting the pattern of electric field
intensity in wavelength λ=400nm, nanojet – which its broad applications in nanoptic and optoelectronic fields
was mentioned – was occurred. Likewise, same computation was held for modeling scattered field from a gold
nanowire with refractive index 𝑛𝑖𝑛 = .44205 + 2.4313i and radius R=150nm in vacuum which is illuminated
by a TM polarized plane wave with operating wavelength λ=547nm. Then, scattering cross section of the
relevant nanowire was calculated and depicted as a function of wavelength interval 200nm to 1000nm. After
analyzing and comparing scattered Hz intensity graph and scattering cross section data, it was achieved that LSP
was excited in the corresponding gold nanowire in wavelength λ=547nm while both of analytical Mie theory
and semi-analytical GMT reached to the same result.
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